Small Inductive Dimension of Topological Spaces

FORMALIZED MATHEMATICS Vol. 17, No. 3, Pages 207–212, 2009

Karol Pąk Institute of Computer Science University of Białystok Poland

Summary. We present the concept and basic properties of the Menger- Urysohn small inductive dimension of topological spaces according to the books [7]. Namely, the paper includes the formalization of main theorems from Sections 1.1 and 1.2.

MML identiﬁer: TOPDIM 1, version: 7.11.04 4.130.1076

The terminology and notation used here are introduced in the following articles: [17], [8], [15], [5], [16], [6], [18], [14], [1], [2], [3], [13], [11], [9], [12], [19], [20], [10], and [4].

1. Preliminaries

For simplicity, we adopt the following rules: T , T1, T2 denote topological spaces, A, B denote subsets of T , F denotes a subset of T A, G, G1, G2 denote families of subsets of T , U, W denote open subsets of T A, p denotes a point of T A, n denotes a natural number, and I denotes an integer. One can prove the following propositions: (1) Fr(B ∩ A) ⊆ Fr B ∩ A.

(2) T is a T4 space if and only if for all closed subsets A, B of T such that A misses B there exist open subsets U, W of T such that A ⊆ U and B ⊆ W and U misses W . Let us consider T . The sequence of ind of T yields a sequence of subsets of 2the carrier of T and is deﬁned by the conditions (Def. 1).

c 2009 University of Białystok 207 ISSN 1426–2630(p), 1898-9934(e) 208 karol pąk

(Def. 1)(i) (The sequence of ind of T )(0) = {∅T }, and (ii) A ∈ (the sequence of ind of T )(n + 1) iff A ∈ (the sequence of ind of T )(n) or for all p, U such that p ∈ U there exists W such that p ∈ W and W ⊆ U and Fr W ∈ (the sequence of ind of T )(n). Let us consider T . Note that the sequence of ind of T is ascending. We now state the proposition (3) For every F such that F = B holds F ∈ (the sequence of ind of T A)(n) iff B ∈ (the sequence of ind of T )(n). Let us consider T , A. We say that A has finite small inductive dimension if and only if: (Def. 2) There exists n such that A ∈ (the sequence of ind of T )(n). Let us consider T , A. We introduce A is finite-ind as a synonym of A has finite small inductive dimension. Let us consider T , G. We say that G has finite small inductive dimension if and only if: (Def. 3) There exists n such that G ⊆ (the sequence of ind of T )(n). Let us consider T , G. We introduce G is finite-ind as a synonym of G has finite small inductive dimension. The following proposition is true (4) If A ∈ G and G is finite-ind, then A is finite-ind. Let us consider T . One can check the following observations: ∗ every subset of T which is finite is also finite-ind, ∗ there exists a subset of T which is finite-ind, ∗ every family of subsets of T which is empty is also finite-ind, and ∗ there exists a family of subsets of T which is non empty and finite-ind. Let T be a non empty topological space. One can check that there exists a subset of T which is non empty and finite-ind. Let us consider T . We say that T has finite small inductive dimension if and only if:

(Def. 4) ΩT has finite small inductive dimension. Let us consider T . We introduce T is finite-ind as a synonym of T has finite small inductive dimension. One can verify that every topological space which is empty is also finite-ind. Let X be a set. Note that {X}top is finite-ind. One can check that there exists a topological space which is non empty and finite-ind. In the sequel A1 is a finite-ind subset of T and T3 is a finite-ind topological space. small inductive dimension of topological . . . 209

2. Small Inductive Dimension

Let us consider T and let us consider A. Let us assume that A is ﬁnite-ind. The functor ind A yields an integer and is deﬁned as follows: (Def. 5) A ∈ (the sequence of ind of T )(ind A + 1) and A/∈ (the sequence of ind of T )(ind A). We now state two propositions:

(5) −1 ≤ ind A1.

(6) ind A1 = −1 iﬀ A1 is empty. Let T be a non empty topological space and let A be a non empty ﬁnite-ind subset of T . Observe that ind A is natural. The following three propositions are true:

(7) ind A1 ≤ n − 1 iff A1 ∈ (the sequence of ind of T )(n). (8) For every finite subset A of T holds ind A < A. (9) ind A1 ≤ n if and only if for every point p of T A1 and for every open subset U of T A1 such that p ∈ U there exists an open subset W of T A1 such that p ∈ W and W ⊆ U and Fr W is finite-ind and ind Fr W ≤ n − 1. Let us consider T and let us consider G. Let us assume that G is finite-ind. The functor ind G yielding an integer is defined by the conditions (Def. 6). (Def. 6)(i) G ⊆ (the sequence of ind of T )(ind G + 1), (ii) −1 ≤ ind G, and (iii) for every integer i such that −1 ≤ i and G ⊆ (the sequence of ind of T )(i + 1) holds ind G ≤ i. The following propositions are true:

(10) ind G = −1 and G is finite-ind iff G ⊆ {∅T }. (11) G is finite-ind and ind G ≤ I iff −1 ≤ I and for every A such that A ∈ G holds A is finite-ind and ind A ≤ I.

(12) If G1 is ﬁnite-ind and G2 ⊆ G1, then G2 is ﬁnite-ind and ind G2 ≤ ind G1.

Let us consider T and let G1, G2 be ﬁnite-ind families of subsets of T . Observe that G1 ∪ G2 is ﬁnite-ind. The following proposition is true

(13) If G is finite-ind and G1 is finite-ind and ind G ≤ I and ind G1 ≤ I, then ind(G ∪ G1) ≤ I. Let us consider T . The functor ind T yields an integer and is defined as follows:

(Def. 7) ind T = ind(ΩT ). Let T be a non empty ﬁnite-ind topological space. One can verify that ind T is natural. The following three propositions are true: 210 karol pąk

(14) For every non empty set X holds ind({X}top) = 0. (15) Given n such that let p be a point of T and U be an open subset of T . Suppose p ∈ U. Then there exists an open subset W of T such that p ∈ W and W ⊆ U and Fr W is ﬁnite-ind and ind Fr W ≤ n − 1. Then T is ﬁnite-ind.

(16) ind T3 ≤ n if and only if for every point p of T3 and for every open subset U of T3 such that p ∈ U there exists an open subset W of T3 such that p ∈ W and W ⊆ U and Fr W is ﬁnite-ind and ind Fr W ≤ n − 1.

3. Monotonicity of the Small Inductive Dimension

Let us consider T3. Observe that every subset of T3 is ﬁnite-ind. Let us consider T , A1. Note that T A1 is ﬁnite-ind. One can prove the following propositions:

(17) ind(T A1) = ind A1. (18) If T A is finite-ind, then A is finite-ind. (19) If A ⊆ A1, then A is finite-ind and ind A ≤ ind A1.

(20) For every subset A of T3 holds ind A ≤ ind T3. (21) If F = B and B is finite-ind, then F is finite-ind and ind F = ind B. (22) If F = B and F is finite-ind, then B is finite-ind and ind F = ind B.

(23) Let T be a non empty topological space. Suppose T is a T3 space. Then T is ﬁnite-ind and ind T ≤ n if and only if for every closed subset A of T and for every point p of T such that p∈ / A there exists a subset L of T such that L separates {p}, A and L is ﬁnite-ind and ind L ≤ n − 1.

(24) If T1 and T2 are homeomorphic, then T1 is finite-ind iff T2 is finite-ind.

(25) If T1 and T2 are homeomorphic and T1 is ﬁnite-ind, then ind T1 = ind T2.

(26) Let A2 be a subset of T1 and A3 be a subset of T2. Suppose A2 and A3 are homeomorphic. Then A2 is ﬁnite-ind if and only if A3 is ﬁnite-ind.

(27) Let A2 be a subset of T1 and A3 be a subset of T2. If A2 and A3 are homeomorphic and A2 is ﬁnite-ind, then ind A2 = ind A3.

(28) If T1×T2 is finite-ind, then T2×T1 is finite-ind and ind(T1×T2) = ind(T2× T1). (29) For every family G3 of subsets of T A such that G3 is finite-ind and G3 = G holds G is finite-ind and ind G = ind G3. (30) For every family G3 of subsets of T A such that G is finite-ind and G3 = G holds G3 is finite-ind and ind G = ind G3. small inductive dimension of topological . . . 211

4. Basic Properties 0-dimensional Topological Spaces

Next we state several propositions:

(31) T is ﬁnite-ind and ind T ≤ n if and only if there exists a basis B1 of T such that for every A such that A ∈ B1 holds Fr A is ﬁnite-ind and ind Fr A ≤ n − 1. (32) Let given T . Suppose that

(i) T is a T1 space, and (ii) for all closed subsets A, B of T such that A misses B there exist closed 0 0 0 0 0 0 0 subsets A , B of T such that A misses B and A ∪ B = ΩT and A ⊆ A and B ⊆ B0. Then T is ﬁnite-ind and ind T ≤ 0. (33) Let X be a set and f be a sequence of subsets of X. Then there exists a sequence g of subsets of X such that (i) S rng f = S rng g, (ii) for all natural numbers i, j such that i 6= j holds g(i) misses g(j), and

(iii) for every n there exists a finite family f1 of subsets of X such that S f1 = {f(i); i ranges over elements of N: i < n} and g(n) = f(n) \ f1. (34) Let given T . Suppose T is finite-ind and ind T ≤ 0 and T is Lindelöf. Let A, B be closed subsets of T . Suppose A misses B. Then there exist 0 0 0 0 0 0 closed subsets A , B of T such that A misses B and A ∪ B = ΩT and A ⊆ A0 and B ⊆ B0.

(35) Let given T . Suppose T is a T1 space and Lindel¨of.Then T is ﬁnite-ind and ind T ≤ 0 if and only if for all closed subsets A, B of T such that A misses B holds ∅T separates A, B. (36) Let given T . Suppose that

(i) T is a T4 space, a T1 space, and Lindelöf,and (ii) there exists a family F of subsets of T such that F is closed, a cover of T , countable, and finite-ind and ind F ≤ 0. Then T is finite-ind and ind T ≤ 0.

In the sequel T4 is a metrizable topological space. We now state four propositions:

(37) Let A, B be closed subsets of T4. Suppose A misses B. Let N1 be a ﬁnite- ind subset of T4. Suppose ind N1 ≤ 0 and T4N1 is second-countable. Then there exists a subset L of T4 such that L separates A, B and L misses N1. (38) Let N1 be a subset of T4. Suppose T4N1 is second-countable. Then N1 is ﬁnite-ind and ind N1 ≤ 0 if and only if for every point p of T4 and for every open subset U of T4 such that p ∈ U there exists an open subset W of T4 such that p ∈ W and W ⊆ U and N1 misses Fr W. 212 karol pąk

(39) Let N1 be a subset of T4. Suppose T4N1 is second-countable. Then N1 is finite-ind and ind N1 ≤ 0 if and only if there exists a basis B of T4 such that for every subset A of T4 such that A ∈ B holds N1 misses Fr A. (40) Let N1, A be subsets of T4. Suppose T4N1 is second-countable and N1 is finite-ind and A is finite-ind and ind N1 ≤ 0. Then A ∪ N1 is finite-ind and ind(A ∪ N1) ≤ ind A + 1.

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Received June 29, 2009